Let \(T\) denote a commutative ring with unity, and let \(R\) be a \(T\)-algebra. Let \({\mathcal D}(R)=\{x\in R\mid abx=axbx\) and \(xab=xaxb\) \(\forall a,b\in R\}\). We say \(R\) is \(SD\)-generated if \(R\) is generated as a \(T\)-submodule by a subsemigroup of \(({\mathcal D}(R),\cdot)\).
In this paper prime and subdirectly irreducible \(SD\)-generated algebras are described. The following result is proved: Theorem 3.2. Let \(0\neq R\) be a prime \(T\)-algebra. (1) If \(|{\mathcal D}(R)|>1\), then \(R\) has a unity and \({\mathcal D}(R)=\{0,1\}\). (2) If \(R\) is \(SD\)-generated, then \({\mathcal D}(R)=\{0,1\}\), \(R=T1\), \(R\) is a \(T\)-homomorphic image of \(T\), \(R\) is an integral domain, and each \(T\)-submodule of \(R\) is an ideal of the \(T\)-algebra \(R\).
Various corollaries are established, for example if \(R\) is a nonzero prime, \(SD\)-generated \(T\)-algebra and \(T\) is a field, then \(R\simeq T\). If \(T\) is a p.i.d., then \(R\) is isomorphic to either \(T\), or a field \(T/\langle p\rangle\), where \(p\) is a prime in \(T\). If \(R\) is an \(SD\)-generated \(T\)-algebra, then every prime (semiprime) ideal of \(R\) is completely prime (completely semiprime). Moreover, the prime radical of \(R\) coincides with the set of nilpotent elements of \(R\). If \(R\) is left or right primitive, then \(R\) is a field.
The following result concerning subdirectly irreducible \(T\)-algebras is proved: Theorem 3.10. Let \(R\) be \(SD\)-generated. If \(R\) is subdirectly irreducible, with heart \(H\), then exactly one of the following holds: (1) \(RH\neq 0\), \(HR\neq 0\), \(R=T1\), and \(R\) is a homomorphic image of \(T\) (and is hence commutative). (2) \(RH=0=HR\) and \(H\) is a simple \(T\)-module. (3) \(RH\neq 0\), \(HR =0\), \(R\) has a left unity, and \(H\) is a simple \(T\)-module. (4) \(RH=0\), \(HR\neq 0\), \(R\) has a right unity, and \(H\) is a simple \(T\)-module. Various corollaries of this result are established.
For the entire collection see [Zbl 0889.00019].